Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#392109 (order 3072) · Inn(G) = C2×#67 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #488 |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(6,10)(13,18)(14,21)(15,23)(17,24)(19,20)(25,26);(4,6,7,10)(5,8,9,11)(12,14)(13,17)(15,22)(16,21)(18,26)(19,24)(20,25)(23,27);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,19,16,20,23,18)(14,17,22,24,21,25,27,26)Polycyclic presentation:
2^6,3|a^2=e,b^2=d,e^2=f,c^a=cde,g^a=g^2,c^b=cdf,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #67543, #67547, #80516, … |
| maximal normal subgroup of | #67543, #67547, #80516, … |
| maximal quotient of | #69542, #69547, #69554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 615);